Master’s Degree Thesis Mechanical Engineering
Influence Analysis of
Coupling between
Tension and Torque in
Single Armoured Cables
Joacim Malm
Blekinge Institute of Technology, Karlskrona, Sweden 2016
Supervisors: Ansel Berghuvud, BTH
Andreas Tyrberg, ABB
Influence Analysis of Response
Coupling between Tension and
Torque in Helically Single
Armoured Cables
Joacim Malm
Blekinge Institute of Technology
Department of Mechanical Engineering
Karlskrona, Sweden
2016
Following thesis submitted for completion of Master of Science in Mechanical
Engineering with emphasis on Structural Mechanics at Blekinge Institute of
Technology, Karlskrona, Sweden.
in collaboration with
Abstract
When single armoured cables are under tension they will start to twist in
direction depending on the lay angle of the armour wires on the cable. The
cables start to twist due to the induced torque that appear within the cable and
this induced torque can in the worst-case cause loops and kinks on the cable.
These negative consequences are the subject to exploration by implementing a
known analytical solution developed by Lanteigne (1985), into an external
function in OrcaFlex which is a finite element software program capable of
solving 3D non-linear arbitrarily deflections. It is specifically designed to be
used for doing static and dynamic analysis on marine and offshore structures.
Different cable installation scenarios cases have been created and some of these
are validated through experimental testing at the testing facility at ABB,
Karlskrona. The results from the different cases shows a good representation of
the real twist and rotation of the cable. However, results also conclude that the
induced torque that appear within the cable was not possible to be correctly
represented by applying external torque from an external function. These
results are important for further research within the area and they help creating
a greater understanding of torque related problems with single armoured cable
under tension.
Keywords:
Single armoured cables, External function, Induced torque, Loops and kinks.
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Sammanfattning
När kablar som är enkelarmerade utsätts för drag eller tryck så kommer de börja
vrida sig i riktning beroende på läggningsvinkel på armeringstrådarna.
Kablarna börjar vrida på sig på grund av det inducerade vridmomentet som
uppkommer i kabeln, i värsta fall så kommer detta inducerade vridmoment leda
till uppbyggnaden av kinkar och loopar på kabeln. Dessa negativa
konsekvenser är studerade genom att implementera en känd analytisk lösning
utvecklad av Lanteigne (1985), som en extern funktion i OrcaFlex. OrcaFlex är
ett program som nyttjar finita element metoden för att lösa 3D olinjära
godtyckliga nedböjningar. Det är speciellt skapat för att användas för att göra
statiska och dynamiska analyser av marina strukturer.
Olika fall för kabelinstallationsscenarios has skapats och några av dessa
valideras även genom experimentell provning i det mekaniska laboratoriet på
ABB, Karlskrona. Dessa resultat från de olika fallen visar på en bra
representation av den verkliga vridningen samt rotation av kabeln, emellertid
visar resultaten även att det inducerade vridmomentet inte var möjligt att bli
korrekt representerat genom att applicera externt vridmoment genom en extern
funktion. Dessa resultat är viktiga för fortsatt forskning inom området och de
hjälper till att skapa en bättre förståelse för vridmomentsrelaterade problem
med kablar som är enkelarmerade under drag.
Nyckelord:
Enkelarmeradekablar, Externa funktioner, Inducerat vridmoment, Kinkar och
loopar.
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Preface
As this thesis states, at least for the near future, the end of my academic career
I would like to take this moment to express my gratitude towards those that
guided me throughout this amazing experience. This thesis work conducted at
Blekinge Institute of Technology and performed at ABB HVC in Karlskrona,
would not have been possible without the invaluable supervision and guidance
from Ansel Berghuvud, BTH, Andreas Tyrberg, ABB, and Johan Hedlund,
ABB.
I thank you all deeply.
Joacim J. Malm
Karlskrona, May 24th, 2016.
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Nomenclature
Notations
Symbol
Description
Axial strain (m)
Axial stiffness (N)
Axial stiffness with fixed ends (N)
Axial stiffness for free rotating end (N)
Bending angle (rad)
Bending moment (Nm)
Coordinate along cable (m)
Cross section area of the cable core (m2)
⌇
Displacement vector
[ ]
Elongation (m)
Infinitesimal small strain energy (J)
Inner radius (m)
Lay angle (rad)
Length of strained wire (m)
Length of wire (m)
[
]
Load vector
Normal strain (m)
Normal stress (Pa)
Partial differential
⬇
Planar moment of inertia of the cable core (m4)
,
Polar coordinates (m, rad)
Polar moment of inertia of the cable core (m4)
xi
Pure shear stress (Pa)
Pure shear strain (m)
Rotation angle (rad)
⤇
[ ]
Shear modulus of the cable core (Pa)
Stiffness coefficient for coupled axial torsion
Stiffness matrix
Tension (N)
Torsional stiffness (Nm2)
Torque (Nm)
Torque tension coupling coefficient
Total number of armour wires in the layer (#)
Total internal strain energy (J)
Twist tension coupling coefficient
Variation of external energy (J)
Variation of external work energy (J)
Variation of virtual work (J)
Variation of the internal strain energy (J)
Young’s modulus for armour wire (Pa)
Young’s modulus for cable core (Pa)
…..
…….
Acronym
Description
ASEA
Allmänna Svenska Elektriska Aktiebolaget
ABB
ASEA Brown Boveri Ltd
BBC
Brown Boveri et Cie
CAS
Computer Algebra Calculation
XLPE
Cross-Linked Polyethylene
xii
DOF
Degrees of Freedom
DoFs
Degrees of Freedom System
FE
Finite Element
HVC
High Voltage Cables
CWI
National Research Institute for Mathematics and
Computer Science
MI
Paper Lapped
RAOs
Response Amplitude Operators
…..
…..
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Table of Contents
1
INTRODUCTION ................................................................ 1
2
THEORETICAL FRAMEWORK ...................................... 7
3
METHOD ........................................................................... 13
4
RESULTS ........................................................................... 23
Background
Organization of ABB
Problem
Problem statement
Purpose
Delimitations
Within coupling between tension and torque
Within OrcaFlex
Related research
Modelling
Coupling between tension and torque
Theoretical axial stiffness and simplifications
For free rotating end
For fixed end
Theoretical background for hanging cable
Fixed – Fixed boundaries
Fixed – Free boundaries
Implementation
Programming
Python external function
Simulation
Case 1 – Cable under constant tension
Case 2 – Cable under varying tension
Case 3 – Free hanging cable with seabed interaction
Experimental testing
Test 1
Test 2
From analytical model and external function
Case 1 - Cable under constant tension
Fixed – Fixed boundaries
Fixed – Free boundaries
Free – Free boundaries
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Case 2 - Cable under varying tension
Fixed – Fixed boundaries
Fixed – Free boundaries
Case 3 – Free hanging cable with seabed interaction
Fixed – Fixed boundaries
Fixed – Free boundaries
Analysis of results from experimental test
Test 1
Simulation of result in Test 1
Test 2
Simulation of result in Test 2
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8
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DISCUSSION ...................................................................... 51
Discussion of analytical model and external function
Case 1 - Cable under constant tension
Fixed – Fixed boundaries
Fixed – Free boundaries
Free – Free boundaries
Discussion of hanging cable
Case 2 – Cable under varying tension
Case 3 - Free hanging cable with seabed interaction
Discussion of results from experimental test
Test 1
Test 2
Discussion of results
Comparison of results for a cable with constant tension and
one fixed boundary and one free
Comparison of results for a cable with constant tension and
free boundaries
Discussion of methodology
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CONCLUSIONS ................................................................. 57
RECOMMENDATIONS AND FUTURE WORK ............ 59
REFERENCES ................................................................... 60
Literature
Web pages
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62
APPENDIX A: DERIVATION OF COUPLING BETWEEN
TENSION AND TORQUE
APPENDIX B: DESCRIPTION OF ORCAFLEX
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1 INTRODUCTION
This chapter describes the background, problem statement, purpose and
delimitations within the project.
Background
Organization of ABB
Through the merging of the Swedish company Allmänna Svenska Elektriska
Aktiebolaget (ASEA) and the Swiss company Brown Boveri et Cie (BBC) in
1988, ASEA Brown Boveri Ltd (ABB) was formed. Today this merged
company with headquarter in Zürich employs over 155 000 persons in over 100
countries. ABB is today known as one of the global leaders within
automatization and power technologies. In Sweden ABB has over 9300
employees at over 30 locations and 800 of these is located at HVC (High
Voltage Cables) in Karlskrona. HVC is a subdivision of Power Systems and is
manufacturing high voltage cables for both land and sea (ABB, 2016).
Problem
The principle of helically armoured wires is used in a large variety of different
applications such as suspension bridges, cranes, electromechanical-,
communication- and electrical cables, only to mention a few. ABB are
manufacturing their high voltage cables from the same principle. However, the
single armoured cables they manufacture, which are extensively being used for
shallow water applications such as off-shore wind, are not torque balanced.
Representation of one of these cables can be seen in Figure 1.1. Where the
conductor is made of copper or aluminium and the conductor screening is
extruded. The insulation is either of XLPE (Cross-Linked Polyethylene) or
paper lapped (MI). The screening of the insulation should be semi conductive
so a smooth dielectric surface is created (Worzyk, 2009). The dielectric
insulation must be protected by water ingression and high voltage submarine
cables normally have a sheath made by extruded lead to protect from water
ingression. The most salient part of the cable is the armouring, which is
important for both protection towards surroundings and to have good stability
1
towards tension and torsion. The armouring is often made from galvanized steel
wire, but it happens that copper, aluminium or brass also are being used.
Figure 1.1 Representation of a helically single armoured cable, which is not
balanced against torque
Axial tensional forces appear not only during the installation when the cable is
hanging down towards the seafloor but also from the movement of the vessel
these are referred to as dynamic axial tensional forces (Worzyk, 2009). When
an axial tensile force is applied on this type of cable, a torque is introduced
within the cable, trying to twist and unwind the cable. This induced torque can
potentially result in formation of loops and kinks at the touch down point during
installation as can be seen in Figure 1.2 (Knapp, 1979). If a cable has loops
caught in it and as the tension once again increases the cable may undergo a
2
plastic deformation around the loops and this can further lead to an unwanted
permanent damage on the cable (Coyne, 1990).
Figure 1.2 Loops and kinks created by induced torque (Perkins, 2016).
It is also well known that this induced torque has negative consequences on the
total strength of the cable (Knapp, 1979). The single helically armoured cable
consists of a number of armour wires wrapped around the central core in a
helically structure. The constants as seen in Figure 1.3 that defines the structure
of the cable is the lay angle ( ) which is the slope of the armour wire applied
on the cable and the inner radius ( ) which is the distance from the core centre
to the centre of any armour wire in the layer (Lanteigne, 1985).
3
Figure 1.3 Lay angle and inner radius of a single armoured helically cable
Problem statement
Extend the knowledge on how the relation between tension and torque in
helically single armoured cables can influence the global behaviour of the cable
during different cable installation scenarios.
Purpose
As how it is today, it is not possible in the installation analysis software
OrcaFlex to analyse the induced torque that appears when a helically single
armoured cable is under either negative or positive tension. If it were possible
to create a model where this phenomenon is included, a more reliable picture
of the reality would be achieved in the analysis. The purpose of this work is to
identify and show the influence of the induced torque that appears in helically
single armoured cables. These results are beneficial for both creating a greater
understanding and reliability of the model and identifying possible limitations
and dangers within the model.
4
Delimitations
Within coupling between tension and torque
The analytical model by Lanteigne does not account for displacement in the
radial direction. This displacement is small, compared to the elongation in the
length of the cable and has therefore not been considered in the analysis. All
armoured wires are also considered to have the same radius and material
constants. Within the derivation of the coupling, the conductor is modelled as
a homogeneous structure. Calculations will therefore be done for a uniform core
with armoured wires wound helically around it.
Within OrcaFlex
Delimitations within complex softwares such as OrcaFlex, which is specifically
designed to model various structures in dynamic water, is always necessary. It
is possible to include many different features in the simulations and it is always
a matter of prioritizing. It is possible to include complex phenomena such as
nonlinear wave patterns, nonlinear soil models for seabed interactions, different
wave and current induced forces, independent motion of vessel, and many
more. These more advanced features will not be included in the analysis.
However, many of these advanced features will be left for implementation
within future research. This is due to both the extra computational cost and that
there are not any expectations that these additional features will add any
additional value within the data for the analysis. One example that could be
included is the seabed friction, which is a highly complex and greatly nonlinear
feature. It takes into account things like what deformation of the seabed that
would occur if the cable were to be dragged across it. It is also possible to add
wind and current induced forces within the simulation. However, these features,
as many others will not be included.
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2 THEORETICAL FRAMEWORK
This chapter describes related research within the targeted model and shows
the analytical model that will be used for conducting simulations.
Related research
It is well known that when a helically single armoured cable is put under an
external tension, an internal torque appears that will try to twist the cable.
Therefore, any variation in external tension, either negative or positive will
induce the cable with a change of torque (Knapp, 1979). There have been many
different authors trying to predict the structural response of the cable.
The famous work by research scientist Lanteigne (1985) which is widely being
cited throughout this report shows a theoretical estimation of the response of
helically armoured cables to tension, torsion, and bending. It produces a method
to determine the response for helically armoured cables under static loading
conditions for both intact cables and when an arbitrary number of constituent
wires have failed and will leave the cable unbalanced. It was mainly based on
the work of Knapp (1979) which developed a new stiffness matrix for straight
helically armoured cables considering tension and torsion. Knapp took into
consideration the compression of the central core element, which created
internal geometric nonlinearities. The true pioneer in this area though is Hruska
(1952) which developed methods for determining the mechanical properties of
armoured wire ropes and this method opened up for all following research and
literature around this subject. Raoof and Hobbs (1988) proposed a change to
the work of Knapp (1979), and instead of considering an aluminium core with
steel reinforced cables they chose to use an orthotropic material formulation
and view the armour wires as shells.
Coyne (1990) did an analysis of the formation and elimination of loops in
twisted cables it furthermore investigated what phenomena that would occur
when a fast relaxation of a cable that previously had been under an axial
tensional load was done quickly.
7
Slippage of the wires in the cables during bending was briefly introduced in the
work of Lanteigne but was further developer by Sævik and Gjøsteen (2012) in
a strength analysis modelling of flexible umbilical members for marine
structures.
Modelling
The used model is mainly built upon the previous work of Lanteigne (1985),
however Lanteigne (1985) did not show all details regarding the derivation of
the established equations. This chapter will follow a similar approach but will
show the mathematical steps more clearly and help create a much deeper
understanding of the model. Lanteigne (1985) derives the formulation for a
number of layers of wires but in this case, the derivation is done for one layer
of armour wire.
Coupling between tension and torque
From the derived models (See Appendix A) from Lanteigne (1985), the
following relation is stated:
[ ][ ] = [
]
where ] is the stiffness matrix,
load vector. They are given by:
] is the displacement vector and
(2.1)
] is the
[ ]=
(2.2)
[ ]=
(2.3)
[
]=
where , ,
and
(2.4)
is the applied tension, torque, elongation and twist. The
internal attributes of the stiffness matrix
8
] is given by:
Axial stiffness:
( )+
=
⌇
(2.5)
( ) +⤇
(2.6)
where , , ⌇ , and is the cable armouring properties; number of armouring
wires, Young's modulus of wire, cross section area of wire and the lay angle of
the cables.
and ⌇ is the cable core properties; Young's modulus of the core
and cross section area of the core.
Torsional stiffness:
( )
=
where is the radius from the centre of the core to the centre of the armouring
wire core. ⤇ and is the cable core properties; shear modulus of the core and
the polar moment of inertia of the core.
Stiffness coefficient for coupled axial torsion:
( )
=
( )
(2.7)
By expanding equation 2.1, the coupling between tension and torque is given
by:
=
=
+
+(
(
−
−
)
(2.8)
)
(2.9)
where Equation 2.8 is describing torque with relation to tension and elongation
and Equation 2.9 is describing torque with relation to tension and radial twist.
9
Theoretical axial stiffness and simplifications
For free rotating end
The theoretical axial stiffness at a free rotating end can be calculated by solving
Equation 2.1 with zero torque in the boundary, hence using:
=
−
(2.10)
)
where the theoretical axial stiffness at a free rotating end is given by:
= (
( )
⌇
( )
⌇ −
( )
( ))
( )
)
(2.11)
Further simplifications can be made for free boundaries, where it is impossible
for torque, and this result in the following equation for twist:
(2.12)
where
is considered the twist tension coupling coefficient and it is
calculated as:
=
(2.13)
)
For fixed end
The theoretical axial stiffness at a fixed end can be calculated by solving
Equation 2.1 with zero rotation in the boundary, hence using:
=
)
where the theoretical axial stiffness with fixed ends is given by:
=
( )
(2.14)
(2.15)
⌇
10
Further simplifications can be made for fixed boundaries where it is impossible
for radial twist and this result in the following equation for torque:
(2.16)
where
is considered the torque tension coupling coefficient and it is
calculated as:
=
Theoretical background for hanging cable
(2.17)
From simple mechanics, Roden (1989) describes the distribution for tension,
twist and torque on fixed- free boundaries, and free-free boundaries. Which can
be seen in Figure 2.1 and Figure 2.2.
Fixed – Fixed boundaries
With the cable-ship still at sea and a part of the cable with a rotational restriction
at the boundaries is lowered to the bottom of the sea. The resulting distribution
of tension, twist and torque can be seen in Figure 2.1 for fixed – fixed
boundaries. The tension is changing linearly along the length of the cable and
it is because of the built up in mass. The tensile force creates a torque in the
cable and since the cable is prevented from twisting at the ends, an internal
torque will be generated at the boundaries. This torque will be constant along
the cable (Roden, 1989).
The twist that can be seen in Figure 2.1 can best be described by; at the upper
part, close to the ship, the cable will try to untwist and straighten the armour
resulting in negative twist. At the bottom of the cable where there is not any
tension there will not be any untwisting of the armouring, furthermore the
elastic nature of the cable will resist untwisting and result in a positive twist. It
can be explained better by viewing the cable in two different parts. The part
above the half-depth of the cable will untwist and the part below the half-depth
will be tightened (Roden, 1989).
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Figure 2.1 Distribution of tension, twist and torque of a free hanging cable
with fixed-fixed boundaries (Roden, 1989).
Since the area in the upper part of the twist is equal to the area of the lower part
it can be said that the angle of rotation at the boundaries must be equal to zero.
This is furthermore proven by our basic knowledge of that fixed boundaries are
not possible to rotate.
Fixed – Free boundaries
If the same example as in chapter 2.3.1 were used but with the lower boundary
released. It would cause the cable to unwind and straighten itself.
Figure 2.2 Distribution of tension, twist and torque of a free hanging cable
with one fixed boundary and one as free (Roden, 1989).
Hence the lower boundary at the cable is free to rotate when being lowered
towards the ocean floor it will start to unlay and the twist is depending of the
local tension at each node across the cable. The stretched area inside twist in
Figure 2.2 can be seen as the number of rotations or total angle that the cable
has rotated around its neutral axis (Roden, 1989).
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3 METHOD
In this chapter the implementation of the external function will be presented
and simulations will be described. Furthermore, the equipment and process for
experimental testing will be described.
Implementation
Implementation of the coupling between tension and torque within OrcaFlex is
done by using an external function that will help to insert the wanted torque at
each node on the OrcaFlex model. Further information regarding models within
OrcaFlex and environmental influences can be viewed in Appendix B.
Programming
Python is a completely open source high level programming language, which
was developed by Guido van Rossum at CWI (National Research Institute for
Mathematics and Computer Science) in Netherlands between the late eighties
and the early nineties. Python resemble many other programming languages
such as ABC, C, C++ and UNIX (Python, 2016).
Python external function
The external function is a construction of three classes. What the three classes
all have in common is that they all are constructed by an initialise part and a
calculation part. The initialise part handles the model objects, period, steps and
finds where the applied loads are located on the model. The calculation part
then uses the gathered information from the initialise part to calculate the value
that will be used for each applied load (Python, 2016).
The first class is created to be used for the first fixed or free boundary and
applies the equation for calculating torque from tension and twist as can be seen
in Equation 2.9. What it does is that it identifies which tension and twist that
exists at each node and then uses these found values to calculate the torque and
then applies this torque to the node. The OrcaFlex model is built up by segments
(See Appendix B) and it is the number of created segments that is controlling
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how many applied loads the external function shall create. A rule of thumb is
that with a higher number of segments the model will have an increase of
accuracy this though comes with the cost of extra computational time.
The second class is looking for a gradient of twist and tension between the
nodes. If this gradient exists, a torque will be inserted in the applied load. This
process is fully iterative since the model uses twist within the equation and with
a higher twist the higher torque will be inserted. When the result reaches
convergence, the value is accepted and the process starts again for the next time
step.
The third class operates if there exists a similarity with the boundaries. If both
boundaries are fixed or free this class calculates as the first class with the
exception that the applied torque will have the same formula as with class one
but will be negative.
A number of steps can describe the solving process.
Model is created in
OrcaFlex with a
selected number of
segments
The iterative process
reaches convergence
for each time step
Python script applies
loads at each node at
the segments
Solving process starts
with torque applied on
the model
Python script enter the
external function at
each applied load
A new model is
created for OrcaFlex
where the model has
influence from torque
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The external function starts at = −10s. This is before the dynamics is applied
on the model which is at = 0 this is so the torque can be staggered and that
the model will find convergence at each step. This is done mainly because it is
preferable to increase the applied torque in a controlled manner rather than
applying it all at once. If the torque is applied all at once, it would be seen as
an impulse for the simulation and it would introduce unwanted motions
(Dingeman, 1997).
Simulation
The external function is applied at an arbitrary amount of nodes and this will
cause the cable to rotate. If the external function is disabled, no twist will occur.
Simulations for three different cases where the first and second case will help
to establish the reliability of the created external function and during the third
case the dynamic movement, wave pattern and integration with the seabed will
be included.
Case 1 – Cable under constant tension
A constant force will be applied on the cable and three different pair of
boundary conditions will be simulated which can be seen in Figure 3.1. First
simulation will show what happens when a cable which is fixed in both
boundaries, is pulled with a constant tension. Second simulation will have one
fixed boundary and one free, and the third simulation within the cable with
constant tension will be for a cable that is fully free to rotate in both boundaries.
15
Figure 3.1 The three cable under constant tension simulation scenarios
Case 2 – Cable under varying tension
To receive a varying tensional load, the cable can be positioned in such a way
that it is hanging longitudinal in the air, as can be seen in Figure 3.2. This will
simulate the condition when the cable is hanging freely from the vessel out in
the sea. The varying tension is generated from the weight of the free hanging
cable. Simulations will be done for a hanging cable with both ends fixed and
one simulation for what will happen when the bottom boundary is free to rotate,
which can be seen in Figure 3.2.
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Figure 3.2 The two cable under varying tension simulation scenarios
Case 3 – Free hanging cable with seabed interaction
By including the hanging cable which was presented in case two and increasing
the length of the cable so it will be in full contact with the seabed. This will
give a more realistic observation of a cable installation scenario which can be
seen in Figure 3.3.
The influence of airy waves and elastic linear seabed model (See Appendix B)
will be included in the simulation.
Simulations will be done for both when the cable is fixed in both boundaries
and when the bottom part can rotate freely as the upper part is fixed.
17
Figure 3.3 Free hanging cable scenario of case 3 (image adapted from
original (Perkins, 2016)).
Experimental testing
Two full scale tensile tests have been performed on two different independent
cables at the testing facility at HVC ABB, Karlskrona. To measure the results,
elongation and rotation sensors were applied on the cable, which can be seen in
Figure 3.4.
From the experimental testing it is possible to identify the twist, elongation,
force and most of all the true axial stiffness of the cable.
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Figure 3.4 Type of elongation and rotation sensors that were used during the
experimental testing.
The elongation is measured with help of the mounted sensors on the cable
where elongation is given by:
=
where
is the stressed length and
is the unstressed length.
(3.1)
Test 1
A single armoured, three core, AC cable with the length of 45.7 m which can
be seen in Figure 3.5 were pulled in three cycles with a load between 100kN to
a total of 500kN. The cable is fully attached in one end and can rotate freely in
the other. In the first load cycle the cable will straighten and the measurements
from the first load cycle will not be used for analysis. The second and third
cycle has a much higher credibility and it is these that will be presented.
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Figure 3.5 Single armoured, three core, AC cable and moulding device
To help measure the elongation and rotation, angle and elongation sensors were
applied on the cable with a given distance in between which can be seen in
Figure 3.6. Four angle sensors were used to measure the rotation of the cable,
equally distributed over a distance of 25m from the fixed boundary.
Figure 3.6 Placing for angle and elongation sensors
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Test 2
A single armoured DC cable with the length of 50m, which can be seen in
Figure 3.7, were pulled with a constant tension of 300kN. The cable can rotate
freely in both ends of the cable. To help measure the elongation and rotation,
angle and elongation sensors where applied on the cable which can be seen in
Figure 3.8.
Figure 3.7 The experimental setup of the tensile test of a single armoured, DC
cable
21
Figure 3.8 Placing for angle and elongation sensors
Chinese fingers which can be seen in Figure 3.9 where attached with swivels at
the boundaries to create the wanted free spinning condition. Three load cycles
are applied, where the load change from 50kN to 300kN. The first cycle is done
so the cable can straighten itself and these measuring results will not be used
for analysis. The second and third cycle has a much higher credibility and it is
these that further will be presented.
Figure 3.9 Shows the Chinese finger that is attached to enable a rotational
motion
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4 RESULTS
In this chapter the results from the analytical solution will be compared with
the computational simulations. Results from the hanging cable will be
compared with the known mechanics and furthermore an analysis will be
presented on the results from the experimental testing.
From analytical model and external function
Data parameters used in the analysis with the numerical model is the data
parameters calculated by the analytical model.
Case 1 - Cable under constant tension
In the cable under constant tension simulations were performed on a cable with
the length of 50m and the constant effective tension for results in chapter 4.1.1.1
and 4.1.1.3 is 300kN and for results in chapter 4.1.1.2 the constant effective
tension is set to 400kN.
Fixed – Fixed boundaries
From Equation 2.10 it can be calculated and shown that all nodes over the cable
with constant tension should have a constant internal torque. The cable should
be squeezed tighter and no twist should be possible. A representation of the
scenarios is given in the element case of a fixed-fixed beam in Figure 4.1.
23
Figure 4.1 Element case of a helically single armoured cable with both
boundaries fixed
A visual representation of the computational simulation of a fixed – fixed cable
can be seen in Figure 4.2. A better view of twist and external torque can be
observed in Figures 4.3 and 4.4.
Figure 4.2 Twist and elongation of a cable with fixed ends at a constant
tension of 300kN
From Figure 4.3 it is shown that if there is a constant tension with two fixed
ends it will give a twist that will approach zero the more the cable is stretched.
The external torque, which can be seen in Figure 4.4 states that the applied
24
torque needs be more frequent at the boundaries because they will try to twist
and get loose from the boundaries.
Figure 4.3 Twist of a cable with fixed ends at a constant tension of 300kN
Figure 4.4 External torque on a cable with fixed ends at a constant tension of
300kN
25
Fixed – Free boundaries
From Equation 2.12 it can be observed that the cable should not have any
internal torque since the cable can rotate freely and all possible internal torque
is transferred into twist. A representation of the scenarios is given in the element
case of a fixed-free beam in Figure 4.5.
Figure 4.5 Element case of a helically single armoured cable with one fixed
boundary and one free
By inserting the same data parameters that was used in the simulation with the
tensional force 400kN into the analytical Equation 2.12 the analytical twist is
calculated to 0.56
.
A visual representation of the computational simulation of a fixed – free cable
can be seen in Figure 4.6. A better view of twist and external torque can be
observed in Figure 4.7 and Figure 4.8.
26
Figure 4.6 Twist and elongation of a cable with one free and one fixed end at
a constant tension of 400kN
From Figure 4.7 it can be seen that if the cable has a constant tension with one
fixed and one free end it will give a constant twist over the cable. The external
torque, which can be seen in Figure 4.8 states that it will be an applied torque
as constant over the cable except at the free boundary since it cannot take up
any external torque.
Figure 4.7 Twist of a cable with one free and one fixed end at a constant
tension of 400kN
27
Figure 4.8 External torque on a cable with one free and one fixed end at a
constant tension of 400kN
The computational simulation gives the resulting twist at 400kN of 0.58
which can be seen in Figure 4.7 and the external torque of 8.2
which can
be seen in Figure 4.8.
Free – Free boundaries
From Equation 2.12 it can be observed that the cable should not have any
internal torque since the cable can rotate freely and all possible internal torque
is transferred into twist. A representation of the scenarios is given in the element
case of a fixed-free beam in Figure 4.9.
28
Figure 4.9 Element case of a helically single armoured cable with both
boundaries free
By inserting the same data parameters as was used in the simulation with the
tensional force 300kN into the analytical Equation 2.12 the analytical twist is
calculated to −1.71
/ .
A visual representation of the computational simulation of a free – free cable
can be seen in Figure 4.10. A better view of twist and external can be observed
in Figures 4.11 and 4.12.
Figure 4.10 Twist and elongation of a cable with free ends at a constant tension of 300kN
29
If there is a constant tension on a cable with one fixed and one free end it will
give a constant twist, which can be seen in Figure 4.11. The external torque,
which can be seen in Figure 4.12 states that the applied torque will be constant
over the cable except at the free boundaries since a free boundary cannot take
up any external torque.
Figure 4.11 Twist of a cable with free ends at a constant tension of 300kN
Figure 4.12 External torque on a cable with free ends at a constant tension of
300kN
30
The computational simulation gives the result for twist at 300kN of
−1.65
which can be seen in Figure 4.11 and the external torque of
−2.5
which can be seen in Figure 4.12.
Case 2 - Cable under varying tension
In the cable under varying tension simulations, a cable with the length of 400m
is used and the tensional force is varying from 1900kN to 0kN. The tensional
force is exaggerated to further show on the resulting shape of the hanging cable.
Since in this case it is only possible to compare with the known shape that is
explained further in chapter 2.3: Theoretical background for hanging cable.
Fixed – Fixed boundaries
A visual representation of the computational simulation of a hanging fixed –
fixed cable can be seen in Figure 4.13. A better view of twist and external torque
can be observed in Figure 4.14 and Figure 4.15.
31
Figure 4.13 Twist and rotation of a free hanging cable with fixed ends
If there is a varying tension on a cable with both ends as fixed, it will give a
linear twist, which can be seen in Figure 4.14 and the external torque, which
can be seen in Figure 4.15 states that the applied torque will be linear over the
cable except at the fixed where the external torque will spike.
32
Figure 4.14 Twist of a free hanging cable with fixed ends
Figure 4.15 External torque of a free hanging cable with fixed ends
Since there is a linear twist over the cable, it will give a second order
polynomial rotation with absolute maximum of the rotation at the middle of the
cable and zero rotation at the boundaries of the cable.
33
Fixed – Free boundaries
A visual representation of the computational simulation of a hanging fixed –
free cable can be seen in Figure 4.16. A better view of twist and external torque
can be observed in Figure 4.17 and Figure 4.18.
Figure 4.16 Twist and rotation of a free hanging cable with one free and one
fixed end
If there is a varying tension on a cable with one fixed end and one free it will
give a linear twist which can be seen in Figure 4.17 and the external torque
which can be seen in Figure 4.18 states that the applied torque will be linear
over the cable.
34
Figure 4.17 Twist of a free hanging cable with one free and one fixed end
Figure 4.18 External torque of a free hanging cable with one free and one
fixed end
Since there is a linear twist over the cable it will give a second order polynomial
rotation with absolute maximum of the rotation at the end of the cable and zero
rotation at the fixed boundary.
35
Case 3 – Free hanging cable with seabed
interaction
In the simulations for the free hanging cable with seabed interaction a cable
with the length of 150m is used. For the static case when the waves are still the
water depth is 100m. When the wave influence is added this depth will vary
throughout the whole simulation.
Normal friction (See Appendix B) within the seabed is added to the simulations.
Fixed – Fixed boundaries
A visual representation of the computational simulation of a hanging fixed –
fixed cable with seabed interaction can be seen in Figure 4.19 and the tensional
force along the cable is varying from 120kN to 40kN, which can be seen in
Figure 4.20.
Figure 4.19 Twist and rotation of a free hanging cable with both boundaries
fixed
36
Figure 4.20 Effective tension at the specific arc length of a free hanging cable
with both boundaries fixed
A better view of twist, external torque and rotation of the cable can be observed
in Figure 4.21, Figure 4.22 and Figure 4.23.
Figure 4.21 Twist of a free hanging cable with both boundaries fixed
37
Figure 4.22 External torque of a free hanging cable with both boundaries
fixed
Figure 4.23 Rotation of a free hanging cable with both boundaries fixed
From both the visual representation of the cable in Figure 4.19 and the rotation
plot in Figure 4.23 it can be seen that the maximum rotation is in the middle of
the free hanging part of the cable. Since the cable is fixed at the boundaries, no
rotation can occur there.
38
Fixed – Free boundaries
A visual representation of the computational simulation of a hanging fixed –
free cable with seabed interaction can be seen in Figure 4.24 and the tensional
force is varying from 80kN to 0kN which can be seen in Figure 4.25.
Figure 4.24 Twist and rotation of a free hanging cable with one free and one
fixed end
From Figure 4.24 and 4.25 it is also shown that in the valley of the wave the
cable will be under compression.
39
A better view of twist, external torque and rotation of the cable can be observed
in Figure 4.35, Figure 4.36 and Figure 4.37.
Figure 4.25 Effective tension at the specific arc length of a free hanging cable
with one boundary fixed and the other one free
Figure 4.26 Twist of a free hanging cable with one free and one fixed end
40
Figure 4.27 External torque on a free hanging cable with one free and one
fixed end
To receive the wanted twist of the cable the external torque is applied. What is
similar with the first and second case is that when there is a linear altering
tension there will be a linear altering twist. From Figure 4.28, which is the plot
of the rotation of the cable, it can be seen that the cable almost rotates 360
degrees.
Figure 4.28 Rotation of a free hanging cable with one free and one fixed end
41
Analysis of results from experimental test
Test 1
Rotation [deg]
From the experimental testing of a cable with one end free and the other one as
fixed, it can be seen that the smallest rotation is at the fixed boundary and it is
increasing linearly towards the free end, which can be seen in Figure 4.29. The
rotation is also measured when the cable is fully stretched and when it is under
the highest load during the test.
4
3,5
3
2,5
2
1,5
1
0,5
0
0
5
10
15
Length of cable [m]
20
25
30
Figure 4.29 Relative rotation at the position of the applied angle sensors
between 100kN and 500kN.
The value at two points from Figure 4.29 is selected independently and divided
by the length between them to receive the corresponding twist. In Figure 4.30
this method has been done for five independent pairs.
42
0,3
Twist [deg/m]
0,25
0,2
0,15
0,1
0,05
0
0
1
2
3
4
5
6
Measuring relations [#]
Figure 4.30 Twist from applied angle sensors on the cable between 100kN
and 500kN.
For a cable with constant twist, it also has a linear relation for rotation. In this
case, the experimental data shown in Figure 4.29 shows that there cannot be
any rotation at the fixed end at the cable and the maximum rotation occurs at
the free end of the cable.
The elongation is calculated from the known original length and the new
stressed length is plotted from cycle two and three in Figure 4.31 and Figure
4.32 against the corresponding pulling force.
600
y = 528213x + 66,996
Force [kN]
500
400
300
200
100
0
0
0,0002
0,0004
0,0006
0,0008
0,001
Elongation [-]
Figure 4.31 The relation between force and strain in cycle 2 to determine the
axial stiffness
43
600
y = 531854x + 75,788
Force [kN]
500
400
300
200
100
0
0
0,0002
0,0004
0,0006
Elongation [-]
0,0008
0,001
Figure 4.32 The relation between force and strain in cycle 3 to determine the
axial stiffness
The anomaly shape at the beginning and the ending of the measurements at
the force strain diagrams is due to the trouble with correct synchronization of
the force and elongation sensors. However, these errors are so small so they
do not make an impact on the ending slope of the curve.
Based on the following cable data inserted into Equation 2.11:
Conductor cross section, lay angle of cores, and elastic modulus of
conductor
Armour wire cross section, number of wires, lay angle, and elastic
modulus of wires
the stiffness parameters which can be seen in Table 4.1 were calculated.
Table 4.1 Axial stiffness for fixed end and free rotating end for the AC-cable
Cable parameter
Value
589
471
From Figure 4.31 and Figure 4.32 it can be seen that the average measured axial
stiffness is 530MN, which is the slope of the curve. The calculated axial
stiffness for a free rotating end from the analytical equation is 471MN.
44
Simulation of result in Test 1
Cable parameters that were used in the simulation was calculated based on the
analytical Equation 2.11 and can be seen in Table 4.1 and Table 4.2.
Table 4.2 Torsional stiffness and stiffness coefficient for coupled axial torsion
for the AC-cable
Cable parameter
Value
841850
9.99*106
The experimental results from the fixed-free cable shows the relative rotation
from 100kN to 500kN and as the rotation as seen as linear the load for the
simulation where 400kN to better simulate the found experimental results.
From Figure 4.33 it can be seen that if there is a constant tension on a cable
with one fixed and one free end it will give a constant twist of 0.56
over
the cable.
Figure 4.33 Twist of a cable with one free and one fixed end at a constant
tension of 400kN
45
Test 2
From the experimental testing of a cable with both ends free the results indicate
on a close to linear rotation over the cable which can be seen in Figure 4.34.
The smallest rotation is in the middle of the cable and both ends rotate apart
from each other. The rotation is also measured when the cable is fully stretched
and when it is under the highest load during the test.
60
Rotation [Deg]
40
20
0
-20 0
10
20
30
40
50
-40
-60
-80
Length of cable [m]
Figure 4.34 Relative rotation at the position of the applied angle sensors at
300kN.
The value at two points from Figure 4.34 is selected independently and divided
by the length between them to receive the corresponding twist. In Figure 4.35
this method has been done for five independent pairs. The spreading in
amplitude between the values in Figure 4.35 indicate that the measuring
accuracy could be increased.
Twist [Deg/m]
0
-0,5 0
1
2
3
4
5
-1
-1,5
-2
-2,5
-3
-3,5
Measuring relations [#]
Figure 4.35 Twist from applied angle sensors on the cable at 300kN.
46
6
For a cable with constant twist it is also having a linear relation for rotation. In
this case, the experimental data shown in Figure 4.34 shows that the absolute
maximum rotation is biggest at the free boundaries and the middle of the cable
have zero rotation.
The elongation is calculated from the known original length and the new
stressed length and is plotted from cycle two and three in Figure 4.36 and Figure
4.37 against the corresponding pulling force.
Force F [kN]
400
y = 420029x - 77,053
300
200
100
0
0
0,0002
0,0004
0,0006
0,0008
0,001
0,0012
Elongation [-]
Force [kN]
Figure 4.36 The relation between force and strain in cycle 2 to determine the
axial stiffness
350
300
250
200
150
100
50
0
y = 388517x - 156,76
0
0,0002
0,0004
0,0006
0,0008
Elongation [-]
0,001
0,0012
0,0014
Figure 4.37 The relation between force and strain in cycle 3 to determine the
axial stiffness
Based on the following cable data inserted into Equation 2.11:
Conductor cross section, lay angle of cores, and elastic modulus of
conductor
Armour wire cross section, number of wires, lay angle, and elastic
modulus of wires
47
the stiffness parameters which can be seen in Table 4.3 were calculated.
Table 4.3 Axial stiffness for fixed end and free rotating end for the DC-cable
Cable parameter
Value
503
386
From Figure 4.36 and Figure 4.37 it can be seen that the average measured axial
stiffness is 404MN which is the slope of the curve. The calculated axial stiffness
for a free rotating end from the analytical equation is 386MN.
Simulation of result in Test 2
Cable parameters that were used in the simulation was calculated based on the
analytical Equation 2.11 and can be seen in Table 4.3 and Table 4.4.
Table 4.4 Torsional stiffness and stiffness coefficient for coupled axial torsion
for the DC-cable
Cable parameter
Value
88514
-3.26 *106
If there is constant tensional force on a cable with both ends free the simulation
results say that it will have a constant twist of −1.65
which can be seen
in Figure 4.38
48
Figure 4.38 Twist of a cable with free ends at a constant tension of 300kN
49
50
5 DISCUSSION
In this chapter a discussion of the given results from chapter 4 is found.
Discussion of analytical model and external
function
Analytical results compared with the results by implementing the external
function.
Case 1 - Cable under constant tension
Fixed – Fixed boundaries
By restricting rotation at the boundaries for a cable with constant tension, it will
only stretch the cable and no twist will occur. These results are found for both
the elementary case and in the simulated case.
Fixed – Free boundaries
By using the same cable parameters for both in the simulation and within the
analytical calculation it can be seen by doing a straight analytical calculation
that the twist will be calculated to 0.56deg/m and compared this to the twist
within the simulation that was 0.58deg/m. The difference between these
absolute values is 3.5%, which is a considerable accurate value. This
strengthens the belief that the external function is correctly implemented within
the software.
Free – Free boundaries
By using the same cable parameters for both in the simulation and within the
analytical calculation it can be seen by doing a straight analytical calculation
that the twist will be calculated to -1.71deg/m and compared this to the twist
within the simulation that was -1.65deg/m. The difference between these
absolute values is 3.6%, which is a good accuracy. This further strengthens the
belief that the external function is correctly implemented within the software.
51
Discussion of hanging cable
Case 2 – Cable under varying tension
From the research from Roden (1989) which can be viewed in chapter 3.3.2 it
shows a clear representation of the wanted shape of torque and twist on the
cable.
For both the scenarios under case two it is possible to directly compare the
results with the elementary case, they all match for the wanted twist, however
the simulated torque does not correspond. In these cases, it is not possible to
rely on any experimental testing to further validate the method but since the
shape of the twist is similar, the method can in the future be altered only by
adding a constant to alter the influence of the external torque.
It is important to comprehend that the internal net torque must be constant along
the length of the hanging cable for when the cable has both boundaries fixed.
This is since the rotation of the cable is restricted. For simplification, it can be
addressed that if there is any restriction of rotation there will be an internal
torque building up, if there on the other hand is not any restrictions there will
only be twist.
Case 3 - Free hanging cable with seabed interaction
For both scenarios from case three, there is not any earlier study or elementary
cases that gives us a solution to compare with accessible. Instead what needs to
be done is to compare with the found results in the first and second case and
what is possible to conclude is that when the cable has a linear altering tension
it also should have a linear twist, and when there is a constant tension on the
cable it should have a constant twist. These assumptions and realisations can
all be found in the results for both scenarios in case three.
Is it sufficient to exclude the computational heavy external function and instead
replace it with a single twisting moment at the free boundary of the cable?
A simplification to receive the wanted twist is to totally neglect the external
function and instead apply a torque at the free boundary that will give the
52
average twist of the cable. This can be suitable when a fast computational
calculation is wanted since the usage of the external function greatly increases
the time it takes to solve the simulation.
Discussion of results from experimental test
Through the experimental testing two different cables where used to validate
the analytical and external function. It would have been more beneficial to use
the same cable for the two different cases since the exact correlation between
the two cables is not known. For future experimental tests, a specific cable
could be used to give a better clear picture of the results. The acceleration when
stretching the cables was done manually and It would have been beneficial to
try to use different accelerations and compare these to see if the acceleration
has any influence in the result.
Test 1
From the experimental testing of the cable with one boundary free and the other
one free the results state that the twist on the cable is 0.15deg/m and by
comparing this with both the computational simulation and analytical
calculation, which gave the result 0.56deg/m. The results indicate on a big
difference on how free the cable is towards rotation as these results indicate that
the free boundary at the experimental testing might not be fully free to rotate.
Further, it would have been beneficial to increase the number of angle sensors
on the cable to better view the expected linearity of the rotation. However, the
simulation and experimental testing is behaving as expected when the rotation
is increasing towards the free end of the cable.
The analytical axial stiffness for a free rotating AC-cable is calculated to
471MN and from the data from the experimental tests average is calculated to
530MN gives us and relative error of 11%. Which could give us the reason why
there is a lower rotation in the analytical solution compared with the
experimental. The other explanation is that the free rotating end cannot entirely
be seen as a free boundary. This boundary could be more like a semi-free where
there still is some twisting restriction. For a cable with fixed ends the analytical
53
axial stiffness is calculated to 589MN which helps further to indicate that the
free boundary is more semi-fix since the measured axial stiffness is right in
between our calculated axial stiffness for fixed ends and free rotating.
Test 2
From the experimental testing of the cable with both boundaries free the results
state that the twist on the cable is -2.7deg/m and by comparing this with both
the computational simulation and analytical calculation, which gave the result
-1.7deg/m. That gives us an absolute error of 1 degree per meter. The shape of
the rotation from simulation and real life testing is similar except from the slope
of the rotation. The computational simulation can easily be altered by applying
a constant to make it better simulate the reality of the cable. This could be
preferable in some cases where it is interesting to know the exact rotation.
It is highly interesting that the experimental axial stiffness is calculated to
386MN and the measured from the experimental testing is of an average of
404MN, which is a relative error of 5%. This helps us to further believe in the
analytical solution and states that it has a good capacity of estimating the right
result for axial stiffness.
The measured rotation at 50m on the cable is higher than from the analytical
calculations, this conclude that the analytical calculations is underestimating
the result.
Discussion of results
What is interesting in the results is that the shape of the twist and then the
corresponding rotation of the cable match in all scenarios but the torque does
not. This is a big insight and it can be explained further by: Since OrcaFlex
does not have the built in feature of using a single armoured cable a straight
cable is selected, which does not twist when tension is applied. It is possible to
simulate the characteristics between tension, torque and twist by applying an
external torque on the cable and it will give the wanted twist of the cable. What
needs to be stated here is that this applied external torque will not be a
54
representation of the real induced torque that would appear within a single
armoured cable. However, the generated twist this applied torque creates, will.
The wanted twist is found by applying a torque on the cable at an arbitrary
amount of selected nodes. This is however as stated giving us a false torque on
the cable. It is for this moment not possible to make the twist and torque to be
correct at the same time. This is something that has been found that it cannot
be done with the usage of an external function. It somehow needs to be directly
implemented into the simulation software.
Comparison of results for a cable with constant tension and
one fixed boundary and one free
Table 5.1 Comparison between analytical, external function and experimental
testing for a cable with constant tension and one fixed boundary and one free
Method
Twist (deg/m)
Analytical
0.56
External function
0.58
Experimental testing
0.15
From the comparison if the results from the analytical method, external function
and the experimental testing in Table 5.1 it is possible to conclude that the
external function is correctly implemented. However, by comparing it to the
experimental testing the values do not correspond. This indicates on errors
within either the experimental testing or the analytical function.
55
Comparison of results for a cable with constant tension and
free boundaries
Table 5.2 Comparison between analytical, external function and experimental
testing for a cable with constant tension and one fixed boundary and one free
Method
Twist (deg/m)
Analytical
-1.71
External function
-1.65
Experimental testing
-2.7
From the comparison if the results from the analytical method, external function
and the experimental testing in Table 5.2 it is possible to conclude that the
external function is correctly implemented. However, by comparing it to the
experimental testing the values do not correspond. This indicates on errors
within either the experimental testing or the analytical function.
Discussion of methodology
The usage of an external function within OrcaFlex has limitations. What has
been stated before is the fact that with the usage of an external function it is not
possible to receive the true internal torque of the cable. This could maybe be
done by twisting the cable before doing the simulation and this is functional for
static solutions. However, when dynamics is implemented and the tension over
the cable is constantly changed over the time interval the twist also should
change and this method will fail. What needs to be done is to create a model
that will naturally unlay when it is hanging freely. How this is done is outside
of the scope of this work and is a good angle for continuing work.
56
6 CONCLUSIONS
In this chapter conclusions will be presented over the answer provided for the
problem formulation.
The problem statement says that the task was to extend the knowledge between
tension and torque in helically single armoured cables. Throughout an external
function implemented into the simulation software OrcaFlex, the knowledge
has greatly extended for typical cable installation scenarios and ordinary cable
testing scenarios. After this work, it is possible to identify the real twist and
rotation of a given cable for different installation scenarios by introducing an
external torque. This external torque is however not always the real internal
torque that a tensional stressed helically single armoured cable will induce. The
task was to show the influence of the induced torque in the cable, and show
how the cable will react from different applied forces, both dynamically and
statically. This influence has been shown through three different cases and
experimental testing has been done to further validate the results. Through the
usage of an external function in OrcaFlex, the big realization has been made
that at this moment it is not possible to have the real torque and twist at the
same time. A selection needs to be made on which of these aspects that will be
targeted. Main problem for this whole scenario is that simulations are made on
a single armoured helically cable by twisting a torque balanced cable, which
will not induce any torque when tension is applied. The method by using the
external function is a good method to find the twist and rotation of the cable.
57
58
7 RECOMMENDATIONS AND FUTURE
WORK
In the future, this created method can be applied on various cable installation
scenarios, such as when the cables are pulled to shore, since this leads to great
tensional build up inside the cable. However, what is most interesting in that
case is the immense torque that the cable is induced with when it is not torque
balanced, since the usage of an external function within OrcaFlex cannot show
a reliable induced torque some other method needs to be created. Further
research by using the external function and applying an external torque on the
cable can be done when a correct twist and rotation is of interest. What needs
to be stated is that further research needs to be done to fully understand how to
implement or identify the induced torque within the cable. One method is to
use the found twist and create a cable that naturally twist that amount the first
developed method gave in result. What it all comes down to is that it would be
beneficial to try to create a more realistic cable that would truly rotate when it
is hanging freely. This is something that cannot be done with the usage of an
external function and is something that OrcaFlex needs to implement directly
into their software.
To further try to validate the results in the experimental test 1, where the
assumption was that the free boundary was not entirely free towards rotation a
model could be created with the usage of OrcaFlex. This model should have a
semi-fix boundary to see if it better matches the found axial stiffness from test
1 in the experimental testing.
59
60
8 REFERENCES
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tension and torsion,” International Journal for Numerical Methods in Engineering, vol. 14, no. 4,
pp. 515–529, 1979.
Langhaar, H. L. (1989) “Principles of Virtual Work and Stationary Potential Energy”, Strain
Energy Methods in Applied Mechanics, pp.214-223.
Lanteigne, J. (1985) “Theoretical Estimation of the Response of Helically Armored Cables to
Tension, Torsion, and Bending,” Journal of Applied Mechanics, vol 52, pp. 423–432.
Rerkins, N. (2011) “Thesis Report on Dynamic Fatigue Loads on Composite Downlines in
Offshore Service”, Airborne Oil & Gas.
61
Rabe, P. P. (2015) “Thesis Report on Dynamic Fatigue Loads on Composite Downlines in
Offshore Service”, Airborne Oil & Gas.
Raoof, M., Hobbs, R. E. (1988) “Analysis of multi-layered structural strands,” Journal of
Engineering Mechanics, vol 114, no. 7, pp. 1166-1182.
Ragab, A-R., Bayoumi, S. E. A. (1998) “Fundamentals and Applications”, Engineering
Solid Mechanics, pp.270-271.
Roden, C. E. (1989) “Submarine Cable Mechanics and Recommended Laying Procedures”,
pp.103-106.
Sævik, S., Gjøsteen, Ø. J. K. (2012) Strength Analysis Modelling of Flexible Umbilical
Members for Marina Structures, Department of Marine Technology.
Søreide, T. H. (1986) “Collapse Analysis of Framed Offshore Structures”, The Norwegian
Institute of Technology.
Worzyk, T. (2009) “Design, Installation, Repair and Environmental Aspects,” Submarine
Power Cables, pp. 27-50.
Web pages
ABB., Kort om ABB. Retrieved from: http://new.abb.com/se/om-abb/kort [18 February 2016]
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http://www.abb.se/cawp/seabb361/dd5ce102d6e2635ac1256b880042aee5.apx [18 February
2016]
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[2 Mars 2016]
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http://www.tutorialspoint.com/python/python_overview.htm [22 February 2016]
62
APPENDIX A:
A.1
DERIVATION OF COUPLING BETWEEN
TENSION AND TORQUE
Determination of axial strain in wire
Figure A.1 Strain of an arbitrary selected wire m
We know that the axial strain which can be seen in Figure A.1 is given by:
=
We also see from Figure A.1 that the following connections exist:
=
( )
63
(A.1)
(A.2)
=
( +
) +(
+
( ))
where
is derived using the Pythagoras theorem and
trigonometry.
(A.3)
using simple
Using Equation A.2 and A.3 with Equation A.1 the axial strain of the m:th wire
becomes:
=
(1 +
( )+(
)
( )
( )) − 1
+
(A.4)
A simplification is done by linearizing Equation A.4 by neglecting the influence
of the second order binomial strain expansions. Hence using the relations,
1+
Δ
=2
and
(
( )
Δ
Δ
+
( )) = 2
Δ
( )
( )
( )
( )
leads to the equation,
+1 =
2
( )
+2
(A.5)
By further expanding left hand side of Equation A.5 and using the relation,
(1 +
) =2
We arrive at the following equation for calculating linear axial strain:
=
( )
+
( )
( )
64
(A.6)
A.2
Determination of axial strain in wire due to
bending
Furthermore, we will continue the linear analysis by applying the relation of
bending. By introducing bending to a small element of the wire by the bent
length of small curvature of
.
As can be seen in Figure A.2 we can also observe that between point 1 and 2
we have an axial elongation behaviour and between point 2 and 3 the wire will
have an axial compression behaviour. At point 2 we have the wire crossing the
neutral axis.
These assumptions are done by following the route of an ordinary wire, what
should be mentioned is the fact that the friction within the wire is what is
helping keeping the cross section intact after bending it (Knapp, 1979).
Figure A.2 Bending in cable shows both elongation and compression
As seen in Figure A.2 we can consider the elongation of the wire at a distance
( ) from the neutral axis. By applying following condition taking the
65
banding into account and expanding the axial strain, hence
Equation A.6 it can be rewritten into:
=
+
( )
( )
(A.7)
As can be seen in Figure A.3 we have the following relations
(A.8)
=
( )
( )
+
( )
+
( )
This equation is not balanced towards the neutral axis. We need to define how
the equation changes due to varying position of z. This is done by finding an
expression for the height of the position of the wire and then adding the
influence of z. First we consider the equation to be able to function for any wire
around the cable.
⨇ (0) =
0≤
=
≤
( )) +
,0 ≤
(
)
≤ 2 , and
and where K is the total number of wires in the layer
(A.9)
Then we also need to add the relation of increment of position z so that we can
have a mean axial strain of the m:th wire in the layer.
⨇ ( )=
(
+
) +
(
66
)
(A.10)
Figure A.3 Cross section of configuration at z=0
We can also from Figure A.2 observe that
=
Where
=
( )
(A.11)
is the lay length of the layer.
67
( ) = ⨇ ( ) inserting in
Combine the Equation A.10 and A.11 by letting
Equation A.7 we can have a formulation for describing the mean axial strain of
the wires. Hence,
+
=
( )
( )
+
( )
+
( )
( )
+
(
)
(A.12)
This equation is giving the axial strain of the m:th wire in the layer of a cable.
Where the equation has an axial deformation, rotational strain and curvature
part of the wire.
A.3
Energy methods
We know from the first law of thermodynamics also known as the law of
conservation of energy that: “energy cannot be created or destroyed in a
chemical reaction”. What this means is that energy can neither be destroyed nor
created. It can only be changed or transferred from one form to another. The
internal strain energy, which is created when a system is put under load, will be
the same as the external potential energy hence;
For elastic materials, we have the principle of minimum potential energy and it
follows directly from the principle of virtual work. It defines as,
=
+
(A.13)
=
−
(A.14)
Where
is the variation of the internal strain energy and
defines as the
variation of the external work energy of the applied or external loads as −
so we have the following equation for minimum potential energy:
If we consider the external loads as conservative, excluding for example any
frictional manner within the system we have that the overall total potential
energy should be equal to zero. Hence:
68
=
(A.15)
This states that the internal strain energy is equal to the external work energy
when no energy losses occur (Ragab and Bayoumi, 1998).
Furthermore, if the loads are removed the created strain energy will restore the
body to its original structure if the material still is within the elastic region. If
the material exceeds this region and crosses the point of yield the material will
start to plasticise and will not go back to its original structure (Sævik and
Gjøsteen, 2012).
A.3.1 Benefits with energy based derivations
If we have a conservative structural system, there are numerous benefits of
making the system energy based.
Since energy based derivations is mainly using differentiation this method is
straightforward and it is less likely to get errors prone to the operation. If we
have a more complicated energy gradient derivation a computer program such
as CAS can be used to reduce the possibility of making errors in the calculation
process.
Energy based derivations is not depending on the selection of coordinate system
due to the invariance properties of the simplified transformation of residual
equations. If the coordinates were to change, we can simply change the DOF in
the partial derivatives.
Since we performed the linearization, the tangent stiffness matrix is guaranteed
to be symmetric. This is seen as an advantage for example when investigating
eigenvalues
One major concern when solving with energy based derivations is the loss of
stability of the system but this can have helped by introducing the singular
stiffness criterion. If we have a non-conservative system, we might have to
perform a dynamic criterion test which can increase the stability but with the
cost of extra computational effort.
69
A.3.1.1 Internal strain energy
Consider a small cubic element in the wire, which carries a set of loads as
shown below in Figure A.4.
Figure A.4 A small cubic element with a set of loads applied on it
If we consider d to be the infinitesimal small strain energy of this small cubic
element it can be written as, (Gavin, 2015):
=
where , , and
pure shear strain.
+
+
+
(A.16)
is the normal stress, normal strain, pure shear stress and
By integrating the infinitesimal small strain energy over the volume the total
strain energy becomes:
=
∫
=
∫{ } { }
+
+
(A.17)
By rewriting the number of stresses and strains into vectors we have:
(A.18)
Expanding stresses further with Equation A.19 known as the Hook’s law,
70
=
The total internal strain energy is then given by:
∫{ } { }
=
(A.19)
(A.20)
Where the total internal strain energy is bounded by both strain energies from
the cable core and the sum of all wires in the layer.
+ ∑
=
(A.21)
By controlling the change of displacements of the wires by (. . ) the variation
of the internal strain energy of the wires can be written as:
=
where, [12],
=
where
+ ∑
(A.22)
∫{ }
{ }
is the volume of element
and can be rewritten as,
=
( )
∫ ∫ { }
(A.23)
( )
{ }
where { } is the axial strain increment for
The variation of the internal strain energy of the cable core
combining the applied loads, (Søride, 1989):
,
=
⌇ ∫
( )
(A.24)
is given by
(A.25)
71
,
,Ἄ
=
⤇∫
=
⬇ ∫
Ἄ
( )
(A.26)
(A.27)
Ἄ
where
and
, ,
, ,
,Ἄ , Ἄ is the variation of the internal strain
energy and strain due to axial deformation, rotational strain and flexion. The
specific strain values can be written as:
(A.28)
=
Ἄ
(A.29)
=
(A.30)
=
By solving the integration, the formulation for the total variation of the internal
strain energy in the cable core is given by, (Hjaji and Mohareb, 2013):
=
⌇
+
⤇
+
(A.31)
⬇
Where
is the Young’s modulus, ⌇ is the cross section area, is the polar
moments of inertia, ⤇ is the shear modulus and ⬇ is the planar moments of
inertia, of the cable core.
By substituting Equation A.12 into Equation A.24 and solving it, and including
Equation A.31 the internal strain energy variation for number of wires for
Equation A.22 the solution is given by, (Lanteigne, 1985):
+
=
⌇
(
( )
+
+
⤇
( )
72
+
( )
⬇
+
)
+
(
+
(
+
( )
( )
( )
+
+
Where the constants
+
and
( )
=
( )
)
+ 2)
2
( )
( )
+
−
( )
( )
(A.32)
are as follows:
−
( )
=
(
( )
+
+
( )
( )
A.3.1.2 External stationary potential energy
(A.33)
(A.34)
We consider the elastic structure of the cable which are under the influence of
a set of distributed and in some cases discrete loads which can be seen in Figure
A.5. The stationary potential energy is equal to zero gives:
= ∑
+∑
+∑
Where is the applied pulling tension,
bending moment.
73
(A.35)
is the twisting torque and
is the
Figure A.5 Elastic structure with influence of distributed loads
It follows that the potential energy can be rewritten according the principle of
virtual work as:
= ∑
)+∑
(
)+∑
(
(
(A.36)
)
We have from Equation A.15 that the internal energy is equal to the external
energy this gives:
= ∑
)+∑
(
(
)+∑
(
(A.37)
)
Since the strain energy is constructed as function of displacement it can be
rewritten according to Castiglioni’s first theorem, it gives:
= ∑
ℌ(
)
(
)+∑
)
ℌ(
(
)+∑
ℌ(
)
(
)(A.38)
By using the relation from Castiglioni’s first theorem in Equation A.38 and
placing it into Equation A.35 we can show that
74
∑
(ℌ(
) (
)
−
)+∑
) (
)=0
(ℌ(
)
− ) (
)+∑
(ℌ(
)
−
(A.39)
This relation from Equation A.39 can only be satisfied for all possible axial
deformations, rotational strains and flexion if:
ℌ(
)
ℌ(
)
ℌ(
)
=
(A.40)
=
(A.42)
(A.41)
=
Furthermore, if we consider only to have = 1, = 1 and = 1 number of
influencing loads we have the internal strain energy from external stationary
potential energy to be, (Langhaar, 1989):
A.4
=
(
)+
(
)+
(
)
Coupling between tension and torque
(A.43)
By setting the internal strain energy variation from Equation A.32 equal to the
derived internal strain energy variation trough the external stationary potential
energy Equation A.43 we have obtained the following relation:
[ ][ ] = [
]
(A.44)
where [ ] is the stiffness matrix, [ ] is the displacement vector and [
load vector. They are given by:
] is the
[ ]=
(A.45)
[ ]=
(A.46)
75
[
]=
where the internal attributes of the stiffness matrix [ ] is given by:
( )+
=
=
=
=
(
( )
( )
=∑
=
( )+∑
and
( )
)+
( )
( )
⌇
Where the constants
=
( ) +⤇
)
(
=∑
⌇
( )
+
+
( )
( )
If we have a symmetric cable (no breakage of strands), we have:
∑
∑
−
−
(A.49)
(A.50)
(A.51)
(A.53)
are as follows:
−
( )
(A.48)
(A.52)
−
( )
⬇
(A.47)
( )
+
( )
+
=0
=0
(A.54)
(A.55)
(A.56)
(A.57)
If Equation A.52 and A.53 is set to zero some changes is made for the internal
attributes of the stiffness matrix [ ] to ,
and
. New values for them
are:
76
(
=
)
( )+
=0
⬇
=0
This will change the structure of the stiffness matrix [ ] into:
[ ]=
0
0
0
0
(A.58)
(A.59)
(A.60)
(A.61)
The bending moment of the cable is no longer coupled with either tension or
torque. This means that if we are exposing the cable for axial tensile load we
will not have any bending flexion, only a radial twist.
77
APPENDIX B:
B.1
DESCRIPTION OF ORCAFLEX
Description of system
OrcaFlex is a finite element software programs capable of solving 3D nonlinear arbitrarily deflections. It is specifically designed to be used for doing
static and dynamic analysis on marine and offshore structures.
OrcaFlex can be used to solve numerous complex problems however the usual
problem as with any engineering tool applies that if the operator is
inexperienced seemingly correct results can be achieved even though the
carried out analysis is badly constructed. However, under the usage of a skilful
operator OrcaFlex can be used to create any type of offshore structure from
lumped mass elements.
From Figure B.1 we can observe the OrcaFlex line model that shows that the
weight, mass and buoyancy are lumped to the nodes. The torsion and axial
properties of the segments in the actual pipe are expressed as straight rods that
are attached to a node at each end. Each node in the discretised model is
expressed by combining two half segments, this is always the case with an
exception of the nodes at the boundaries.
78
Figure B.1 OrcaFlex Line Model Based on Lumped Mass Method (Karegar,
2013).
This lumped catenary structure can describe a system of flexible marine risers,
towed systems or umbilical cables. The system can then be evaluated against
an offshore environment with conditions for such as waves, current loads and
externally influencing motions such as wind only to mention a few (Hjaji and
Mohareb, 2013).
With the usage of external functions analysis of pre, intra and post processing
of the results can be done. For this thesis mainly the intra processing will be
used since this is what will insert the torque relation continuously as the process
is running.
The same goes for all engineering software programs that a mathematical
model is created to attempt to describe the reality. However, to reduce the
computational cost simplifications is always made and by using the lumped
79
mass approach OrcaFlex can noticeably reduce the computational cost of the
mathematical calculations.
B.2
Numerical calculations
B.2.1 Statics in OrcaFlex
Before a proper dynamic analysis of the structure can be done the static
calculation part must be done. This is done to determine the expected shape of
the structure and the static equilibrium for each line and buoy element. The
static solution can be calculated with the use of the quick method, catenary
method or even a specific line shape. However, the most commonly used
method is the catenary and it can calculate both hydrostatic and hydrodynamic
forces such as: gravity, friction due to seabed, buoyancy, drag from wind or
current and axial elasticity.
To reach static equilibrium OrcaFlex perform a series of iterative steps:
1. At the start of the process the coordinates for the buoys and vessels
will be calculated by the given data, also a free hanging configuration
is calculated for each line according to the specific model.
This first step of the process is important for further calculations since it uses a
FE model to determine forces, moments and movements and this FE model is
depending on correct coordinates for calculating deviations.
2. Calculate the static equilibrium for each line. To do this the line ends
should be set as fixed. Out of balance forces on each line will be
determined.
The second step will estimate a new position for the entire body when
determining the out of balance forces.
3. Iterate the first and second step until out of balance forces converge
within specified tolerance range.
80
However, these static solutions neglect the influence of bending stiffness and
rotational stiffness but since this is only for the static equilibrium it still can be
seen as sufficient.
B.2.2 Dynamics in OrcaFlex
After the static equilibrium is reached, the next stage is to solve the dynamic
part. Which is created by time simulations for predetermined time increments.
This implies that solutions for external forces that are depending on time can
be achieved. The principal difference between the static and dynamic part is
that OrcaFlex introduces the Newton’s equation of motion for the dynamic part
and it is given by:
( , )+ ( , )+ ( )=
( , , )
(B.1)
where ( , ), ( , ), ( ) and ( , , ) is the system inertia load, system
damping load, system stiffness load, and the external load. The depending
variables , , and is stated as position, acceleration, velocity, and time.
In the beginning of the dynamic solution, the orientation of all objects and their
initial position will be collected from the static solution to solve the influencing
moment and forces for each object and it can be solved using either the explicit
integration scheme or the implicit integration scheme. Whereas the explicit
integration scheme solves the simulation using the forward Euler integration
with a constant time step to get the acceleration at each node. The solution at
the end of each time increment are given by:
( +
( +
where
) = ( )+
)= ( )+
(B.2)
( )
(B.3)
( )
is the time increment.
This process is then repeated for each time increment until all nodes are again
known.
81
The explicit integration scheme generally takes a longer time to solve since it
involves very short time increments. However, for the implicit integration
scheme OrcaFlex uses a Generalised- integration scheme which is an iterative
solution process that solves the equation of motion at the end of each time
increment. Generally, the implicit scheme is stable for a bigger time increment,
hence it often means that the implicit scheme solves faster. This can though be
a problem for simulations that have a very high non linearity for example an
impact simulation (Cribbs, 2010).
B.3
Global analyses
B.3.1 Finite element modelling
The modelling of the cable will be done using line type elements. Parameters
that will be included are:
Outer diameter of the cable [ ]
Mass per unit length [ / ]
Bending and torsional stiffness [
Axial stiffness [ ]
]
B.3.2 Motion of vessels
The motion of the vessels in OrcaFlex is controlled by RAOs (Response
Amplitude Operators) which relates the movement of the vessel to the exposing
waves. The depending variables for the RAOs is the size and mass of the
vessels, and size and periodicity of the waves. The motion of the RAOs, which
can be seen in Figure B.2, can be split into two categories; rotation and
translation. Where the rotationally part is controlled by pitch, yaw and roll, and
translation is controlled by sway, surge and heave. Where the roll, pitch and
heave motion generally is considered the most important parameters.
82
Figure B.2 Positive motion as rotation and translation for the six DOF for the
vessel (Cribbs, 2010).
The displacement of the RAOs is defined in vectors such as amplitude and
phase. Where the phase vector is defined as the relation between the wave and
the motion of the vessel. The amplitude vector is defined as the relation between
the amplitude of the wave and the amplitude of the motion of the vessel (Cribbs,
2010).
B.3.3 Modelling environment
B.3.3.1 Wave models
OrcaFlex has a variation of both regular and irregular wave patterns to use.
Regular wave patterns that exists are the Cnoidal, Stokes 5th order and Airy
Waves, which can all be seen in Figure B.3.
83
Figure B.3 Regular wave patterns that exists in OrcaFlex
The Cnoidal wave pattern is best suited for shallow water and want to model
long waves. It is a periodic wave pattern, which generally has a sharp crest with
long wide troughs. This wave pattern is preferable to use when the ratio
between the depth and wave length is less than 1/8 (Karegar, 2013). The
equation for Cnoidal wave pattern is given by, (Drazin, 1977):
(
)
+⨇
(2 ( )
)
(B.4)
where is the horizontal surface elevation depending on position and time
, is the trough elevation, ⨇ is the wave height, is the phase speed, is the
wavelength,
is the one of the selected Jacobi elliptic functions and
) is
the complete elliptic integral of the first kind depending on the shape of the
cnoidal wave.
The Stokes wave pattern is best suited for when we have and want to model in
deep or intermediate waters. A regular nonlinear wave has a lower steepness
and smoother crest compared to the Cnoidal wave. This wave pattern is
preferable to use when the ratio between the depth and wave length is greater
84
than 1/8 (Karegar, 2013). The equation for Stokes wave pattern is given by,
(Dingeman, 1997):
( , )=
where
=
=
ℎ(
(
(
)+
−
(2(
−
))
(B.5)
(B.6)
)
(B.7)
The Airy wave pattern, which is the simplest of the three regular wave patterns,
is a first order sinusoidal linear wave. It mainly consists of one frequency
component to describe the motion of the sinusoidal wave. This pattern is
preferable if a small influence of the wave pattern is wanted and if we have
deep waters and small waves (Karegar 2013). The equation for Airy wave
pattern is given by:
( , )=
(
−
(B.8)
)
Sometimes the regular wave patterns are not sufficient for describing the model,
in those cases a more accurate description of the sea surface is generated by
finding the solution of a Fourier series analysis of sinusoidal waves. The wave
can be made periodic if we repeat a considered limited time then we have a
periodic motion with the given period (Rabe, 2015). Some of these more
accurate descriptions are the irregular wave patterns Ochi-Hubble, ISSC and
JONSWAP. Irregular waves are used mainly to take into account the big
variation of waves in the wave spectrum. It gives the possibility to include the
static benefit of waves of different sizes and it helps avoid resonance
phenomena that can appear with the usage of regular wave patterns.
If we look into the wave spectrum, which is the distribution of energy as a
function of frequency for these patterns, we see that the energy within these
harmonic waves is proportional to the squared amplitude. A plot of frequency
versus energy can therefore show a random wave. An example of this plot can
85
be seen in Figure B.4 where we can see in which frequencies the most wave
energy is concentrated and where there does not exist any wave energy
(Karegar, 2013).
Figure B.4 Spectral shapes and their corresponding time histories (Rabe,
2015).
By doing extensive analysing of the physical process of old wave records there
has been many wave spectra of algebraic form developed. However, the most
famous among these wave spectra is a modified version of the Pierson –
Moskowits wave spectrum known as JONSWAP which has been developed for
the North Atlantic. The mayor difference between them is that over time there
is a difference of the growth of the peak wave height. The formulas are given
by, (DNV, 2007):
( )= ⌇
( )
(
.
)
86
(B.9)
where
⌇
( )=
0.07
0.09
=
Also
‌
⨇
(− (‌ ) )
≤
>
(B.10)
(B.11)
= 1 − 0.287 ( )
( ) is the JONSWAP Spectrum and
(B.12)
( ) is the Pierson – Moskowits
Spectrum. The variables ,
and ⌇ is the non-dimensional peak shape
parameter, spectral width parameter and the normalizing factor.
If the non-dimensional peak shape parameter is set to one (γ = 1) the
JONSWAP spectrum is equal to the Pierson – Moskowits Spectrum.
It is considered that the JONSWAP spectrum should give a reliable model for
3.6 <
(B.13)
<5
where is the wave peak period and ⨇ is the height to the surface of the wave.
It should be mentioned that outside of the given interval in Equation B.10 this
spectrum should be used with caution. With different parameters on γ the
JONSWAP spectrum is shown in Figure B.5.
87
Figure B.5 JONSWAP spectrum where ⨇ = 4
and
= 2 and = 5 (Dingeman, 1997).
= 8 for
= 1,
If the JONSWAP spectrum for waves is selected in OrcaFlex the only variables
that needs to be specified is the non-dimensional peak shape parameter, γ, and
the wake peak period, Tp . OrcaFlex will calculate the other influencing
parameters and generate the irregular wave train that is used in the analysis.
B.3.3.2 Seabed
OrcaFlex has the possibility to use either an elastic seabed model or a nonlinear
soil model. The elastic seabed model is behaving along the normal direction
like an elastic spring, where the seabed stiffness is specified in advance. The
reaction force from the seabed stiffness is given by:
=
where
is the seabed normal stiffness,
the contact area.
88
(B.14)
is the depth of penetration and ⌇ is
B.4
External functions
A possible way to create an integration between OrcaFlex and Python can be
done through an external function, which allows the user to specify a userdefined function through an external variable data source. From a Python script
we can specify these data sources and these external functions is referred to as
Python external functions. Throughout the solving process, OrcaFlex will
continuously call the external function and set the found value as the variable
data. OrcaFlex then recalculates for each time increment and updates the value
for the specified number of nodes.
B.4.1 Applications for the external function
Within OrcaFlex the external function can be used for a wide range of different
applications such as:
6D buoys, vessels and different loads applied on lines.
Angle orientation for wings.
Tension and payout rate for winches.
Reference direction and speed for current.
Bend stiffness for different line types
Primary motion for vessels
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Department of Mechanical Engineering,
Blekinge Institute of Technology, Campus Gräsvik, 371 79 Karlskrona,
Sweden
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